Notes

In the absence of matter, the standard statement of the Einstein equations is that all components of the Ricci tensor—and thus also the Ricci scalar—must be zero (or formally that Rij = 0). But since the vanishing of all components of a tensor must be independent of the coordinates used, it follows that the vacuum Einstein equations are equivalent to the statement RicciTensor . e . e  0 for all timelike unit vectors e—a statement that can readily be applied to networks of the kind I consider in the main text. (A related statement is that the 3D Ricci scalar curvature of all spacelike hypersurfaces must vanish wherever these have vanishing extrinsic curvature.)

Another way to state the Einstein equations—already discussed by David Hilbert in 1915—is as the constraint that the integral of RicciScalar Sqrt[Det[g]] (the so-called Einstein–Hilbert action) be an extremum. (An idealized soap film or other minimal surface extremizes the integral of the intrinsic volume element Sqrt[Det[g]], without a RicciScalar factor.) In the discrete Regge calculus that I mention on page 1054 this variational principle turns out to have a rather simple form.

The Einstein–Hilbert action—and the Einstein equations—can be viewed as having the simplest forms that do not ultimately depend on the choice of coordinates. Higher-order terms—say powers of the Ricci scalar curvature—could well arise from underlying network systems, but would not contribute noticeably except in very high gravitational fields.

Various physical interpretations can be given of the vanishing of the Ricci tensor implied by the ordinary vacuum Einstein equations. Closely related to my discussion of the absence of t2 terms in volume growth for 4D spacetime cones is the statement that if one sets up a small 3D ball of comoving test particles then the volume it defines must have zero first and second derivatives with time.

Below 4D the vanishing of the Ricci tensor immediately implies the vanishing of all components of the Riemann tensor—so that the vacuum Einstein equations force space at least locally to have its ordinary flat form. (Even in 2D there can nevertheless still be non-trivial global topology—for example with flat space having its edges identified as on a torus. In the Euclidean case there were for a long time no non-trivial solutions to the Einstein equations known in any number of dimensions, but in the 1970s examples were found, including large families of Calabi–Yau manifolds.)

In the presence of matter, the typical formal statement of the full Einstein equations is Rμν - R gμν/2  8 π G Tμν/c4, where Tμν is the energy-momentum (stress-energy) tensor for matter and G is the gravitational constant. (An additional so-called cosmological term λ gμν is sometimes added on the right to adjust the effective overall energy density of the universe, and thus its expansion rate. Note that the equation can also be written Rμν = 8 π G (Tμν - 1/2 Tμμ gμν/c4).) The μ, ν component of Tμν gives the flux of the μ component of 4-momentum (whose components are energy and ordinary 3-momentum) in the ν direction. The fact that T00 is energy density implies that for static matter (where E = m c2) the equation is in a sense a minimal extension of Poisson's equation of Newtonian gravity theory. Note that conservation of energy and momentum implies that Tμν must have zero divergence—a result guaranteed in the Einstein equations by the structure of the left-hand side.

In the variational approach to gravity mentioned above, the RicciScalar plays the role of a Lagrangian density for pure gravity—and in the presence of matter the Lagrangian density for matter must be added to it. At a physical level, the full Einstein equations can be interpreted as saying that the volume v of a small ball of comoving test particles satisfies

∂ttv[t]/v[t]  -1/2 (ρ + 3 p)

where ρ is the total energy density and p is the pressure averaged over all space directions.

To solve the full Einstein equations in any particular physical situation requires a knowledge of Tμν—and thus of properties of matter such as the relation between pressure and energy density (equation of state). Quite a few global results about the formation of singularities and the absence of paths looping around in time can nevertheless be obtained just by assuming certain so-called energy conditions for Tμν. (A fairly stringent example is 0 ≤ p ≤ρ/3—and whether this is actually true for non-trivial interacting quantum fields remains unclear.)

In their usual formulation, the Einstein equations are thought of as defining constraints on the structure of 4D spacetime. But at some level they can also be viewed as defining how 3D space evolves with time. And indeed the so-called initial value formulations constructed in the 1960s allow one to start with a 3D metric and various extrinsic curvatures defined for a 3D spacelike hypersurface, and then work out how these change on successive hypersurfaces. But at least in terms of tensors, the equations involved show nothing like the simplicity of the usual 4D Einstein equations. One can potentially view the causal networks that I discuss in the main text as providing another approach to setting up an initial value formulation of the Einstein equations.